Problem: Rewrite as a simplified fraction. ${0.4\overline{7} = {?}}$
Solution: Let $x$ equal the decimal. Set up two equations such that the digits after the decimal point are identical. ${\begin{align*}100x &= 47.7777...\\ 10x &= 4.7777...\end{align*}}$ Subtracting the two equations, we have: ${90x = 43}$ ${x = \dfrac{43}{90}} $